Finite Group Databasebeta

← #13162025

Subgroups — #13162025

Order 1728.

878 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 878 conjugacy classes of subgroups — too many to draw. Only its 127 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3832C211yesmaximal, characteristic
2#11723656C212yesmaximal, complemented
3C3×#63663C212yesmaximal, complemented
4#11722073C212yesmaximal, complemented
5#10591640—31nomaximal
6#10591637—32nomaximal
7C2×C33×Q8C2211yescharacteristic
8C3×#3832—22no—
9C3×#3832—22no—
10C2×C3×C4⋊(C3⋊D3)C412yes—
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×C4×C3⋊D3C412yes—
13C2×C3×C4×C3⋊D3C2211yescharacteristic
14C3×#3748C2214yes—
15#88809—22no—
16#88808—22no—
17C2×#3832D311yescharacteristic
18C2×C3×#239—32no—
19C2×C3×#239—61no—
20#63818—32no—
21D3×#208—34no—
22#63663D312yescomplemented
23C3×#836—32no—
24C3×#836—32no—
25C3×#836—62no—
26#63559—32no—
27#63558—34no—
28C22×C3×C3⋊D3C2×C412yes—
29C22×C3×C3⋊D3C2×C411yescharacteristic
30C33×Q8—22no—
31C2×C33×C4C2×C412yes—
32C2×C33×C4C2311yescharacteristic
33C2×C3×C3⋊Dic3C2×C411yescharacteristic
34C3×C4⋊(C3⋊D3)—24no—
35C3×C4⋊(C3⋊D3)—22no—
36C3×C4×C3⋊D3—22no—
37C3×C4×C3⋊D3—22no—
38C3×C4×C3⋊D3D412yes—
39C3×C4×C3⋊D3D412yes—
40C3×C3⋊(C3⋊C8)—24no—
41#5237—91no—
42#4751—92no—
43C2×C32×Q8—21no—
44C2×C32×Q8—22no—
45C2×C32×Q8—41no—
46C2×C32×Q8C2×D312yes—
47C2×C32×Q8C2×D311yescharacteristic
48#3832—22no—
49#3832—22no—
50C2×C4⋊(C3⋊D3)Dic312yes—
51C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
52C2×C4×C3⋊D3Dic312yes—
53C2×C4×C3⋊D3C2×D311yescharacteristic
54C3×#239—64no—
55C3×#239—64no—
56C3×#239—64no—
57C3×#239—64no—
58C2×C3×D12—34no—
59C2×C3×D12—32no—
60C2×C3×D12—62no—
61C2×C3×D12—61no—
62C2×C3×C4×D3—32no—
63C2×C3×C4×D3—34no—
64C2×C3×C4×D3—62no—
65C2×C3×C4×D3—61no—
66#3748C2×D314yes—
67C3×#208—34no—
68C3×#208—34no—
69C3×#208—64no—
70#3712—62no—
71#3711—64no—
72#3710—62no—
73D3×C3⋊C8—64no—
74C22×C33C22×C411yescharacteristic
75C2×C3×C3⋊D3—22no—
76C2×C3×C3⋊D3—41no—
77C2×C3×C3⋊D3—42no—
78C2×C3×C3⋊D3C22⋊C412yes—
79C33×C4—22no—
80C33×C4C2×D412yes—
81C3×C3⋊Dic3C22⋊C411yescharacteristic
82C3×C3⋊Dic3C22⋊C411yescharacteristic
83C2×C3×#41—91no—
84C2×#239—32no—
85C2×#239—61no—
86C2×#208—92no—
87D3×M16—94no—
88C3×#101—92no—
89#836—32no—
90#836—32no—
91#836—62no—
92#833—92no—
93#823—94no—
94C22×C3⋊D3C2×Dic312yes—
95C22×C3⋊D3C2×Dic311yescharacteristic
96C22×C3×D3—32no—
97C22×C3×D3—34no—
98C22×C3×D3—61no—
99C22×C3×D3—62no—
100C32×Q8—22no—
101C32×Q8—24no—
102C32×Q8—24no—
103C32×Q8—44no—
104C32×Q8—44no—
105C2×C32×C4—24no—
106C2×C32×C4—22no—
107C2×C32×C4—22no—
108C2×C32×C4—21no—
109C2×C32×C4—42no—
110C2×C32×C4—41no—
111C2×C32×C4C4×D314yes—
112C2×C32×C4C2×Dic312yes—
113C2×C32×C4C22×D312yes—
114C2×C32×C4C22×D311yescharacteristic
115C2×C3⋊Dic3C2×Dic311yescharacteristic
116C4⋊(C3⋊D3)—24no—
117C4⋊(C3⋊D3)—22no—
118C4×C3⋊D3—22no—
119C4×C3⋊D3—22no—
120C4×C3⋊D3#6712yes—
121C4×C3⋊D3#6712yes—
122C2×C3×Dic3—32no—
123C2×C3×Dic3—61no—
124C3×D12—68no—
125C3×D12—68no—
126C3×D12—64no—
127C3×D12—64no—
128C3×C4×D3—34no—
129C3×C4×D3—34no—
130C3×C4×D3—62no—
131C3×C4×D3—62no—
132C3×C4×D3—64no—
133C3×C4×D3—64no—
134C3×C4×D3—64no—
135C3×C4×D3—64no—
136C3⋊(C3⋊C8)—24no—
137C3×C3⋊C8—64no—
138C3×C3⋊C8—64no—
139C3×C3⋊C8—124no—
140#413—271noSylow
141C2×C33—21no—
142C2×C33C2×C22⋊C411yescharacteristic
143C3×C3⋊D3—22no—
144C3×C3⋊D3—42no—
145C3×#41—182no—
146C3×#41—182no—
147C2×C3×Q8—21no—
148C2×C3×Q8—22no—
149C2×C3×Q8—41no—
150C2×C3×Q8D3212yes—
151C2×C3×Q8D3211yescharacteristic
152C2×C3×D4—92no—
153C2×C3×D4—91no—
154C22×C3×C4—91no—
155C22×C3×C4—92no—
156#239—64no—
157#239—64no—
158#239—64no—
159#239—64no—
160C2×D12—34no—
161C2×D12—32no—
162C2×D12—62no—
163C2×D12—61no—
164C2×C4×D3—32no—
165C2×C4×D3—34no—
166C2×C4×D3—62no—
167C2×C4×D3—61no—
168C3×M16—94no—
169#208—34no—
170#208—34no—
171#208—64no—
172#208—182no—
173C2×C3⋊C8—182no—
174ΓL(1,𝔽25)—184no—
175D3×C8—184no—
176C22×C32—22no—
177C22×C32—21no—
178C22×C32—41no—
179C22×C32C2×C4×D312yes—
180C22×C32C22×Dic311yescharacteristic
181C2×C3⋊D3—22no—
182C2×C3⋊D3—41no—
183C2×C3⋊D3—42no—
184C2×C3⋊D3#21712yes—
185C2×C3×D3—34no—
186C2×C3×D3—62no—
187C2×C3×D3—64no—
188C2×C3×D3—64no—
189C2×C3×D3—122no—
190C2×C3×D3—122no—
191C2×C3×D3—124no—
192C2×C3×D3—124no—
193C32×C4—22no—
194C32×C4—24no—
195C32×C4—24no—
196C32×C4—24no—
197C32×C4—22no—
198C32×C4—42no—
199C32×C4—44no—
200C32×C4—44no—
201C32×C4D3×D414yes—
202C32×C4C2×#6712yes—
203C3⋊Dic3#21711yescharacteristic
204C3⋊Dic3#21711yescharacteristic
205C3×Dic3—32no—
206C3×Dic3—32no—
207C3×Dic3—61no—
208C3×Dic3—61no—
209C2×#41—91no—
210C2×M16—272no—
211#101—92no—
212#100—272no—
213C33#41311yescharacteristic, complemented, Sylow
214C23×C3—91no—
215C23×C3—92no—
216C22×D3—32no—
217C22×D3—34no—
218C22×D3—61no—
219C22×D3—62no—
220C3×Q8—24no—
221C3×Q8—24no—
222C3×Q8—22no—
223C3×Q8—44no—
224C3×Q8—44no—
225C3×D4—184no—
226C3×D4—182no—
227C2×C3×C4—22no—
228C2×C3×C4—24no—
229C2×C3×C4—22no—
230C2×C3×C4—21no—
231C2×C3×C4—42no—
232C2×C3×C4—41no—
233C2×C3×C4—91no—
234C2×C3×C4—92no—
235C2×C3×C4—92no—
236C2×C3×C4—182no—
237C2×C3×C4—182no—
238C2×C3×C4D3×Dic314yes—
239C2×C3×C4#62412yes—
240C2×C3×C4C2×D3212yes—
241C2×C3×C4C2×D3211yescharacteristic
242C2×Dic3—32no—
243C2×Dic3—61no—
244D12—68no—
245D12—68no—
246D12—64no—
247D12—64no—
248C4×D3—34no—
249C4×D3—34no—
250C4×D3—62no—
251C4×D3—62no—
252C4×D3—64no—
253C4×D3—64no—
254C4×D3—64no—
255C4×D3—64no—
256C3×C8—184no—
257C3⋊C8—64no—
258C3⋊C8—64no—
259C3⋊C8—124no—
260C2×C32—21no—
261C2×C32—22no—
262C2×C32—22no—
263C2×C32—22no—
264C2×C32—21no—
265C2×C32—41no—
266C2×C32—42no—
267C2×C32—42no—
268C2×C32D3×C22⋊C412yes—
269C2×C32C2×#21711yescharacteristic
270C3⋊D3—22no—
271C3⋊D3—42no—
272C3×D3—64no—
273C3×D3—122no—
274C3×D3—124no—
275C3×D3—124no—
276#41—182no—
277#41—182no—
278C2×Q8#111311yescharacteristic
279C2×D4—92no—
280C2×D4—91no—
281C22×C4—91no—
282C22×C4—92no—
283M16—94no—
284M16—542no—
285C2×C8—542no—
286C22×C3—21no—
287C22×C3—22no—
288C22×C3—41no—
289C22×C3—92no—
290C22×C3—182no—
291C22×C3—361no—
292C22×C3—362no—
293C22×C3C2×D3×Dic312yes—
294C22×C3C2×#62411yescharacteristic
295C2×D3—34no—
296C2×D3—62no—
297C2×D3—64no—
298C2×D3—64no—
299C2×D3—122no—
300C2×D3—122no—
301C2×D3—124no—
302C2×D3—124no—
303C3×C4—24no—
304C3×C4—24no—
305C3×C4—22no—
306C3×C4—24no—
307C3×C4—22no—
308C3×C4—44no—
309C3×C4—44no—
310C3×C4—42no—
311C3×C4—91no—
312C3×C4—91no—
313C3×C4D3×#6714yes—
314C3×C4#381112yes—
315Dic3—32no—
316Dic3—32no—
317Dic3—61no—
318Dic3—61no—
319C32—22no—
320C32—21no—
321C32—41no—
322C32#475112yescomplemented
323C32#523711yescharacteristic, complemented
324C23—91no—
325C23—92no—
326Q8—22no—
327D4—184no—
328D4—182no—
329C2×C4—91no—
330C2×C4—92no—
331C2×C4—92no—
332C2×C4—182no—
333C2×C4—182no—
334C2×C4#629412yes—
335C2×C4C2×#111311yescharacteristic
336C8—184no—
337C2×C3—22no—
338C2×C3—22no—
339C2×C3—21no—
340C2×C3—21no—
341C2×C3—22no—
342C2×C3—42no—
343C2×C3—42no—
344C2×C3—41no—
345C2×C3—182no—
346C2×C3—362no—
347C2×C3D3×#21712yes—
348C2×C3#6397411yescharacteristic
349D3—64no—
350D3—122no—
351D3—124no—
352D3—124no—
353C22—92no—
354C22—182no—
355C22—361no—
356C22—362no—
357C22C2×#629411yescharacteristic
358C4—22no—
359C4—91no—
360C4—91no—
361C4#8904912yes—
362C3—21no—
363C3—22no—
364C3—41no—
365C3#1059163712yescomplemented
366C3#1059164011yescharacteristic, complemented
367C2—21no—
368C2—182no—
369C2—362no—
370C2#1172372311yescharacteristic
subgroupG/H
centerC2#11723723
commutatorC22×C33C22×C4
FrattiniC2×C4C2×#1113
FittingC2×C33×Q8C22
socleC2×C33C2×C22⋊C4
prime radicalC2×C3×#3832C2
cosocleC2×C33×C4C23
hypercenterC2×Q8#1113
hypocenterC33#413
LCSG ▷ C22×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×Q8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #11723656C2semidirect
3. C3×#63663C2semidirect
4. #11722073C2semidirect
22. #63663D3semidirect
213. C33#413semidirect
322. C32#4751semidirect
323. C32#5237semidirect
365. C3#10591637semidirect
366. C3#10591640semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×C3×#38322
#117236562
C3×#636632
#117220732
#105916403
#105916373